5 REPRESENTATIONS OF ALGEBRAS WITH INVOLUTION 345 20. Let A be a Banach algebra with unit element <?, and let x*-+x* be a (not necessarily continuous) involution on A. Show that if ||jc|| ^p(x), then A is a star algebra. 5. REPRESENTATIONS OF ALGEBRAS WITH INVOLUTION Let A be an algebra with involution (not necessarily endowed with a norm, and not necessarily having a unit element), and let H be a Hilbert space. A representation^ of A in H is a homomorphism s\-~+ U(s) of A into the algebra St? (H) of continuous endomorphisms of H, such that (15.5.1) U(s*) = (U(s))*. This implies in particular that if s is self-adjoint, then so is U(s). If A has a unit element e, we require in addition that (15.5.2) U(e)=\H. The representation Uis said to be faithful if the homomorphism jh-* U(s) is infective, that is if the relation U(s) • x = 0 for all x e H implies that s = 0. Let H, H' be two Hilbert spaces. A representation s\-+ U(s) of A in H and a representation s\-*U'(s) of A in H' are said to be equivalent if there exists a Hilbert space isomorphism T : H -> H' such that U'(s) = TC/^)!1"1 for all s G A. When H = H', the Hilbert space automorphisms T of H are precisely the unitary elements of the star algebra JSf(H). For T must be invertible and satisfy (T - x \ T - y) = (x \ y) for all x, y in H, so that (T*T * x\y) = (x\y) and therefore T*T= 1H, hence T* = T'1 ; the converse is immediate. Let H1? H2 be two Hilbert spaces, s\-+ U^s) a representation of A in Hl5 and 5-F-> U2(s) a representation of A in H2 . Let H be the Hilbert sum of H! and H2 , so that Hj and H2 are identified with supplementary subspaces of H. If x = Xi + x2 and y = yl + y2 are two elements of H (where xt , yi in H£ for i = 1, 2), then (6.4) If we put U(s) - (x1 + x2) = Ui(s) - xt 4- U2(s) - x2 , it is immediately verified that U(s) E ^f(H) for each s e A, and that jh-> U(s) is a representation of A, called the Hilbert sum of the given representations. t Strictly speaking, a unitary representation. Since we shall not consider other types of representations, we shall suppress the word "unitary," by abuse of language,.